To solve this problem, I will provide the manual solution using the Laplace transform technique and the MATLAB code to solve the given differential equation.

Manual Solution

The given differential equation is:

$m \frac{d^2 x(t)}{dt^2} + c \frac{dx(t)}{dt} + kx(t) = 0$

with initial conditions: $x(0) = x_0, \quad \frac{dx(0)}{dt} = v_0.$

Step 1: Apply Laplace Transform

Taking the Laplace transform of both sides of the equation:

[ \mathcal{L} \left[m \frac{d^2 x(t)}{dt^2} \right] + \mathcal{L} \left[c \frac{dx(t)}{dt} \right] + \mathcal{L} \left[k x(t) \right] = 0 ]

Using the Laplace transform properties: $\mathcal{L} \left[\frac{d^2 x(t)}{dt^2} \right] = s^2 X(s) - s x(0) - \frac{dx(0)}{dt},$ $\mathcal{L} \left[\frac{dx(t)}{dt} \right] = s X(s) - x(0),$ $\mathcal{L} \left[x(t) \right] = X(s).$

Substitute into the equation: [ m \left[s^2 X(s) - s x_0 - v_0 \right] + c \left[s X(s) - x_0 \right] + k X(s) = 0. ]

Step 2: Simplify the Equation

Expand and rearrange: $m s^2 X(s) - m s x_0 - m v_0 + c s X(s) - c x_0 + k X(s) = 0.$

Factorize $X(s)$: $X(s) \left[m s^2 + c s + k \right] = m s x_0 + m v_0 + c x_0.$

Solve for $X(s)$: $X(s) = \frac{m s x_0 + m v_0 + c x_0}{m s^2 + c s + k}.$

Step 3: Partial Fraction Decomposition (if needed)

The denominator is a quadratic polynomial: $m s^2 + c s + k.$

To find the time-domain solution $x(t)$, perform inverse Laplace transform on $X(s)$. Depending on the roots of the characteristic equation: $m s^2 + c s + k = 0,$ we have three cases:

1. Overdamped ($c^2 > 4mk$): Two distinct real roots.
2. Critically damped ($c^2 = 4mk$): Repeated real roots.
3. Underdamped ($c^2 < 4mk$): Complex conjugate roots.

The solution in each case involves standard inverse Laplace transform techniques.

MATLAB Code

Below is MATLAB code for solving the differential equation using the Laplace transform:

matlabCopy code
syms x(t) s X
m = 1; % Mass (example value, replace as needed)
c = 1; % Damping coefficient (example value)
k = 1; % Spring constant (example value)
x0 = 1; % Initial displacement
v0 = 0; % Initial velocity

% Define the differential equation
Dx = diff(x, t);
D2x = diff(x, t, 2);
eqn = m * D2x + c * Dx + k * x == 0;

% Laplace transform the equation
X = laplace(x, t, s);
Dx0 = v0; % Substitute initial conditions
eqn_L = subs(m * s^2 * X - m * s * x0 - m * Dx0 + c * s * X - c * x0 + k * X, ...
             {'x0', 'Dx0'}, {x0, v0});

% Solve for X(s)
X_s = solve(eqn_L == 0, X);

% Inverse Laplace Transform
x_t = ilaplace(X_s, s, t);

disp('Time-domain solution:');
disp(x_t);

% Optional: Numerical solution
t_vals = 0:0.01:10; % Time range
x_num = double(subs(x_t, t, t_vals));
plot(t_vals, x_num);
xlabel('Time (s)');
ylabel('Displacement x(t)');
title('Damped Harmonic Oscillator');
grid on;

Would you like me to:

1. Provide detailed cases for overdamping, underdamping, and critical damping?
2. Expand the MATLAB solution further with comments?
3. Derive specific numerical examples for parameters $m$, $c$, and $k$?
4. Include plots for various damping scenarios in MATLAB?
5. Solve the Laplace transform manually for specific roots?

Tip: Always verify the roots of the characteristic equation before proceeding with inverse Laplace transform to determine the nature of the solution.